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Definition of Functions

If A1 , A2...

Question

If $A_{1}, A_{2}; G_{1}, G_{2}$ and $H_{1}, H_{2}$ be two A.M.s, G.M.s and H.M.s between two quantities a and b then
prove that $A_{1} H_{2} = A_{2} H_{1} = G_{1} G_{2} = a b$

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Solution

$a, A_{1}, A_{2}, b$ are in A.P.....(1)
$a, H_{1}, H_{2}, b$ are in H.P.
$∴ \frac{1}{a}, \frac{1}{H_{1}}, \frac{1}{H_{2}}, \frac{1}{b}$ are in A.P.
Multiply by ab.
$∴ b, \frac{a b}{H_{1}}, \frac{a b}{H_{2}}, a$ are in A.P
Take in reverse order
Or $∴ a, \frac{a b}{H_{2}}, \frac{a b}{H_{a}}, b$ ......(2)
Compare (1) and (2)
$A_{1} = \frac{a b}{H_{2}} a n d A_{2} = \frac{a b}{H_{1}}$
$∴ A_{1} H_{2} = A_{2} H_{1} = a b = G_{1} G_{2}$

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